The mechanical philosophy

Experiments only tell you something if you already have a view about what sort of thing the world is, and by the 1640s a new one was on offer: matter, motion, and nothing else whatever.

The preceding lessons have been full of people measuring: a slope, a magnet, a heart, a column of mercury. None of that yet says what a good explanation looks like when the measuring is done. The answer that spread across Europe in the middle of the century is called the mechanical philosophy, and its central rule is that everything happens by contact, by one piece of matter pushing another. This lesson sets out what that bought and what it forbade, because the prohibition is what makes Newton's achievement so unwelcome to the people best equipped to understand it.

Matter is extension

René Descartes served in two armies, settled in the Dutch Republic, and suppressed his own cosmology, Le Monde, on hearing of Galileo's condemnation in 1633, as the earlier lesson noted. The physics appeared instead inside the Discourse on Method of 1637 and, in full, in the Principles of Philosophy of 1644.

Its starting point is a definition that does enormous work. The essence of body is extension: to be a material thing just is to occupy space, with length, breadth and depth, and nothing else belongs to matter at all. Colour, taste, smell, heat and heaviness are not in bodies; they are effects produced in us by the size, shape, arrangement and motion of the parts of bodies.

Three consequences follow immediately, and they are not modest.

There is no vacuum. If matter is extension, then a region with extension is a region with matter in it, and an empty space is a contradiction in terms. The universe is a plenum, full everywhere, and the space at the top of Torricelli's tube is filled with a subtle matter fine enough to pass through glass. Descartes said so explicitly, and the previous lesson's experiments were, from his point of view, about the mechanical force of the air and not about the void at all.

There is no action at a distance. A body can only affect another by touching it, because there is nothing to a body but extension and nothing it can do but occupy space and move. Any explanation appealing to attraction, sympathy or influence across a gap is not an explanation but a confession, and this is why the whole generation reacted to such talk as a return to the occult qualities of the schools.

And the world is one kind of stuff. The distinction of the very first lesson, a corruptible sublunary region and an unchanging heaven of a fifth element, is not so much refuted as made unstateable. The moon and a millstone are the same sort of thing, differing in the arrangement of their parts, so a physics that works here has to work there.

What it could compute

The programme would be worthless if it were only a metaphysics, and its most impressive product is a number.

Descartes published the sine law of refraction in 1637: light crossing between two media obeys $n_{1} \sin θ_{1} = n_{2} \sin θ_{2}$ , with a constant for each pair of media. Willebrord Snel had the same relation unpublished in 1621 and Ibn Sahl had it in Baghdad in 984, so the priority is tangled, but Descartes is the one who put it into print and used it.

Example. Light strikes water at $45 °$ from the normal. Water has $n = 1.333$ and air effectively $1$ . Where does the ray go, and at what angle does light inside the water fail to get out at all?

Refraction into the water gives $\sin θ_{2} = \sin 45 ° / 1.333 = 0.7071 / 1.333 = 0.5305$ , so $θ_{2} = 32.0 °$ . Going the other way, light inside the water meeting the surface at angle $θ$ escapes only if $1.333 \sin θ \leq 1$ , so the critical angle is $\arcsin (1 / 1.333) = 48.6 °$ ; beyond that it is totally reflected back into the water.

Now you. Why is the second answer needed for the first proper theory of the rainbow?

Answer

Because the ray that makes the bow is reflected inside the drop rather than passing through it. A ray enters the spherical drop, refracts, meets the far surface from the inside, reflects, and refracts again on the way out. Whether that internal reflection happens, and how much light it carries, depends on the angle at the back surface, which is why the escape condition matters. Descartes had all the pieces: a law telling him how much a ray bends at each surface, and the geometry of a sphere to tell him where each ray meets the surface. That is enough to trace any ray through the drop, and the rainbow becomes a calculation rather than a mystery about vapours.

The rainbow, computed

Descartes took a ray entering a spherical drop at height $b$ above the axis, traced it through with the sine law, and worked out its total deviation. He did this numerically, for ten thousand rays, by hand, and found that as the entry point moves across the drop the deviation reaches an extreme and turns back. Near that extreme, many rays leave in almost the same direction, so light piles up there and that direction is bright.

The algebra behind his arithmetic is short. With incidence angle $i$ and refraction angle $r$ , the total deviation of a ray that refracts, reflects once and refracts again is

D = 180 ° + 2 i - 4 r

and this is stationary when

\cos i = \sqrt{\frac{n^{2} - 1}{3}}

Example. Compute the rainbow angle for water with $n = 1.333$ .

$(n^{2} - 1) / 3 = (1.7769 - 1) / 3 = 0.2590$ , so $\cos i = 0.5089$ and $i = 59.4 °$ . Then $\sin r = \sin (59.4 °) / 1.333 = 0.8607 / 1.333 = 0.6457$ , giving $r = 40.2 °$ . The deviation is

D = 180 + 2 (59.4) - 4 (40.2) = 180 + 118.8 - 160.9 = 137.9 °

A deviation of $137.9 °$ means the light comes back towards the observer at $180 - 137.9 = 42.1 °$ from the direction pointing directly away from the sun. The bow is a circle of radius about $42 °$ centred on the shadow of your own head, which is exactly where it is, and anyone can check it with an outstretched hand. This is the first time a common natural appearance was derived, quantitatively and correctly, from a law of matter and a piece of geometry.

Now you. Refractive index varies with colour: red light in water has $n = 1.3318$ and violet $n = 1.3435$ . What does that do to the bow, and what could Descartes not explain?

Answer

Running the same calculation with $n = 1.3318$ gives $42.3 °$ and with $n = 1.3435$ gives $40.6 °$ , so the bow is about $1.7 °$ wide with red on the outside and violet on the inside, which is what is observed. Descartes could not explain the colours themselves. He proposed that the tiny globules of subtle matter making up light are set spinning by the refraction, faster at one edge of the beam than the other, and that the rate of spin is what the eye registers as colour. That is a mechanical story of exactly the required kind and it is wrong. The correct account, that white light is a mixture which the prism separates rather than modifies, and that each colour has its own fixed refrangibility, is Newton's, published in 1672 in his first paper, and it required an experiment Descartes never thought to do: refracting a single separated colour a second time to show that it does not change further.

Vortices

Having banned empty space and action at a distance, Descartes had to explain the solar system with pushes, and his answer was the boldest thing in the book.

The plenum is in motion, and motion in a full space has to be circulatory, since matter can only move by displacing other matter round in a closed loop. The universe is therefore divided into vast whirlpools, each with a star at its centre. Our sun sits at the middle of one, and the planets are carried round in the swirl like straws in an eddy, each settling at the radius where its bulk is in balance with the surrounding matter. Gravity is the same machinery: the fine matter of the vortex, whirling fastest, presses outwards most strongly and so forces coarse bodies such as stones towards the centre.

Give it its due. It explains, effortlessly, three facts that had never been explained: why all the planets go round the same way, why they lie in nearly the same plane, and why gravity acts towards a centre without anything reaching out to grab. It requires no attraction whatever, and it makes the earth's motion inoffensive to the Church, since the earth is at rest relative to the matter immediately around it and merely carried along, an argument Descartes made explicitly after 1633.

Its defect is that it computes nothing. There is no way to derive Kepler's ellipse from a vortex, no way to get the area law, and no way to obtain the three-halves power in the third law. Newton devoted the whole second book of the Principia to showing that fluid vortices cannot produce the observed motions, and that a vortex obeying Kepler's third law would have to have properties no fluid has. That did not settle it quickly: Cartesian physics remained the standard teaching in France until about 1740, and the argument between vortices and attraction ran for two generations.

Mechanism done properly

The strongest work in this tradition was Christiaan Huygens's, and it shows what the programme could achieve when the mathematics was taken as seriously as the metaphysics.

Descartes had stated seven rules of impact, and six of them are wrong, because he took the conserved quantity to be size times speed without regard to direction. Huygens, using the relativity principle from the lesson on motion, worked out the correct rules and showed that what is conserved in an elastic collision is the vector quantity and also the sum of the products of mass with the square of speed. In 1668 the Royal Society asked for solutions to the collision problem, and Huygens, John Wallis and Christopher Wren independently supplied compatible answers.

The pendulum clock is the other achievement, and it is the one that changed daily life. Galileo had noticed that a pendulum's period is nearly independent of the size of the swing. Huygens turned that into a timekeeper in 1656, and analysed it properly in the Horologium Oscillatorium of 1673, where he proved that the exactly isochronous path is a cycloid rather than a circle, derived the relation

T = 2 π \sqrt{\frac{L}{g}}

and published the formula for the outward tendency of a body moving in a circle, proportional to $v^{2} / r$ . That last result is the one Newton needed and did not have to invent.

Example. How long must a pendulum be to beat seconds, meaning a period of two seconds with one swing each way per second?

Rearranging, $L = g T^{2} / 4 π^{2} = 9.81 \times 4 / 39.48 = 0.994$ m. Just under a metre, which is why longcase clocks are the height they are. A pendulum of exactly one metre has a period of $2 π \sqrt{1 / 9.81} = 2.006$ s.

Now you. Mechanical clocks before 1656 kept time to about 15 minutes a day; a good pendulum clock kept it to about 10 seconds. What does that factor buy, and what does the formula above make possible besides timekeeping?

Answer

The improvement is a factor of $900 / 10 = 90$ , which is the largest single jump in the accuracy of any instrument in the century. It makes the timing of astronomical events routine to the second, so that a transit across a telescope's crosshair becomes a precise measurement rather than an estimate, and it makes the longitude problem soluble in principle, since longitude is the difference between local time and the time at a reference meridian, and one minute of time is a quarter of a degree. Huygens's sea trials failed because a pendulum will not keep time on a rolling deck, and the problem was eventually solved with a spring balance by John Harrison in the following century. The formula does something else besides: rearranged as $g = 4 π^{2} L / T^{2}$ , a pendulum is an instrument for measuring gravity. Since $L$ can be measured with a rule and $T$ by counting a thousand swings, $g$ becomes one of the best-determined quantities in physics, and small variations in it with latitude become measurable, which turns out to matter for the shape of the earth.

The prohibition

Step back and the mechanical philosophy is doing two things at once. It is a research programme, extremely productive, which produced the rainbow, the laws of impact, the pendulum clock and the whole idea that qualities reduce to the arrangement of parts. It is also a rule about what counts as an explanation, and that rule is a prohibition: no attraction, no influence across a gap, no property of a body that is not the size, shape or motion of its parts.

Everyone signed it. Galileo invoked it to dismiss the moon's effect on the tides, as the lesson on the trial described. Boyle called himself a corpuscularian and wrote a book against the notion of nature as an agent. Huygens and Leibniz held to it all their lives.

That is the situation into which the Principia arrives, with a force that acts instantaneously between bodies separated by empty space, in inverse proportion to the square of their distance, with no mechanism whatever offered for how it does so. Before that, though, there is a question this lesson has taken for granted: how any of these results reached anyone else at all. Descartes computed the rainbow in a house in the Dutch countryside, Huygens in The Hague, Boyle in Oxford. What made those private activities into a shared body of knowledge is the subject of the next lesson.